Newton-Krylov Methods for the Solution of the k-Eigenvalue Problem in Multigroup Neutronics Calculations
Open Access
- Author:
- Gill, Daniel Fury
- Graduate Program:
- Nuclear Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- October 05, 2009
- Committee Members:
- Yousry Azmy, Dissertation Advisor/Co-Advisor
Yousry Azmy, Committee Chair/Co-Chair
Seungjin Kim, Committee Chair/Co-Chair
Kostadin Nikolov Ivanov, Committee Member
Ludmil Tomov Zikatanov, Committee Member
Robert I Grove, Committee Member
Brian Aviles, Committee Member - Keywords:
- JFNK
Criticality
Neutron transport
eigenvalue - Abstract:
- In this work we propose using Newton's method, specifically the inexact Newton-GMRES formulation, to solve the k-eigenvalue problem in both transport and diffusion neutronics problems. This is achieved by choosing a nonlinear function whose roots are the eigenpairs of the k-eigenvalue calculation and then using Newton's method to solve the nonlinear system. The flexibility resulting from the use of a Krylov subspace method to solve the linear Newton step can be further extended via the use of the Jacobian-Free Newton-Krylov (JFNK) approximation, which requires no knowledge of the system's Jacobian; instead only the ability to evaluate the system residual is necessary. This makes it possible to avoid the computational and memory costs associated with the construction and storage of the Jacobian, resulting in an efficient solution algorithm. Writing the k-eigenvalue problem as a nonlinear function yields a number of formulations, all of which all have the desired roots. For the diffusion approximation, the nonlinear function is written in the form of the generalized eigenvalue problem and a set of preconditioners is developed and applied to the GMRES iterations that are used to solve the linearized Newton problem. Most of the developed methods can be implemented as either Newton-Krylov (NK) methods, where the Jacobian-vector product is formed using the explicitly constructed Jacobian, or via the JFNK approximation, where a finite-difference perturbation is used to approximate the Jacobian-vector product. One particularly effective preconditioning option comprises the use of the standard power iteration to precondition the GMRES iteration on either the right or the left. Preconditioning on the left, denoted JFNK(PI), results in a modified nonlinear system whose implementation only requires the ability to perform a single traditional outer iteration, making this approach relatively simple to wrap around an existing diffusion theory k-eigenvalue problem solver. Other preconditioning options, such as the Incomplete Cholesky decomposition of the within-group diffusion matrix, are also considered. Similar methods were developed for transport theory, cast using an operator notation that greatly simplifies their presentation. All of the nonlinear functions developed are written in terms of a generic fixed-point iteration, with a number of specific fixed-point formulations considered. Each fixed-point scheme represents a viable k-eigenvalue problem solution method, with two of the techniques corresponding to traditionally used iterative schemes. The new methods developed can also be wrapped around existing software in most instances, simplifying the implementation process. Ultimately it is seen that the most effective of the Newton formulations in transport theory is wrapped around a k-eigenvalue formulation that is a very special instance of traditional methods: no upscattering iterations are performed, only one inner iteration completed per outer, using source iteration with the previous outer iterate as the initial guess. This results in a fixed-point iteration that collapses the three possible iteration levels (outer iterations, upscattering iterations, inner iterations) into a single level of iteration. While this formulation of the k-eigenvalue problem converges very slowly if solved as a traditional fixed-point iteration, when coupled with Newton's method it results in very inexpensive Jacobian-vector products. In the Newton approach an extra degree of freedom is introduced by including the eigenvalue as an unknown, meaning an additional relation is necessary to close the system. In the diffusion theory case a normalization condition on the eigenvector was generally used, however in transport theory a number of so-called constraint relations were considered. These fall into two categories: normalization relations and eigenvalue update formulations. It was observed that the most effective of these constraint relations is the fission-rate eigenvalue update, derived directly from the eigenvalue update formula traditionally used to solve the k-eigenvalue problem. Numerical results, including measured performance quantified in number of iterations and execution time, were generated for suites of benchmark problems using the various Newton's Method formulations for the k-eigenvalue problem in both transport and diffusion theories. These results showed that the choice of the perturbation parameter in the JFNK approximation has very little impact on the calculation while the choice of GMRES stopping criterion significantly affects the total cost of the calculation. The size of the GMRES subspace and the maximum number of restarts permitted were also seen to play an important role in the cost of a calculation. While the diffusion formulations showed little sensitivity to the initial guess of the Newton iterations, the transport formulations were seen to potentially diverge or converge to a non-fundamental mode if a poor initial guess was used. This behavior was avoided by performing a single traditional fixed-point iteration prior to initializing Newton's method. Overall, the numerical results showed that the Newton formulation of the k-eigenvalue problem in diffusion theory is competitive with the Chebyshev accelerated power iteration, with the JFNK(PI) formulation generally resulting in quicker execution times. The transport results showed that a number of the Newton formulations developed result in methods that are significantly less computationally expensive than traditional techniques. Results for the well-known C5G7-MOX benchmark problem demonstrate that the Newton approach reduces by a factor of 5 the total number of sweeps necessary to converge the point-wise fission source error to 10E-4. The numerical results generated in this work show that the Newton approach is superior to existing techniques for both transport and diffusion calculations. Furthermore the newly developed methods have been formulated in such a way that they can be implemented as wrappers around existing code sections, requiring little access and modification to existing code, where the computational kernel is typically some variation of a traditional outer iteration. Based on these results, it is plausible that more advanced (via numerical optimization and acceleration techniques) implementations of these approaches could prove to be more efficient than the methods currently used to solve the k-eigenvalue problem in production-level software.